Finite Group Databasebeta

← #36369539

Subgroups — #36369539

Order 22144.

432 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 104 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D173⋊C8C211yesmaximal, characteristic, complemented
2C2×#19149736C212yesmaximal, complemented
3C2×#19149736C211yesmaximal, characteristic, complemented
4#26012357C212yesmaximal, complemented
5C2×#19149641C211yesmaximal, characteristic
6C2×D173⋊C8—24no—
7C2×D173⋊C8—22no—
8C22×#13063443C2212yes—
9C22×#13063443C2212yes—
10#19149736—24no—
11#19149736—22no—
12C22×D692C411yescharacteristic
13C22×C4×D173C2211yescharacteristic
14C22×C4×D173C2211yescharacteristic
15#19149641—22no—
16#19149641—22no—
17C2×#14583803C411yescharacteristic
18C2×#14583803C411yescharacteristic
19C2×C4×Dic173C2211yescharacteristic
20C2×C4⋊C4×C173C411yescharacteristic
21D173⋊C8—28no—
22C2×#13063443—22no—
23C2×#13063443—22no—
24C2×#13063443—22no—
25C2×#13063443—22no—
26C2×#13063443—22no—
27C2×#13063443D414yescomplemented
28C23×D173C2×C411yescharacteristic
29C23×D173C2×C411yescharacteristic
30C2×D692—22no—
31C2×D692—22no—
32C2×D692C814yescomplemented
33C22×Dic173C2×C411yescharacteristic
34C22×Dic173C2311yescharacteristic
35C2×C4×D173—22no—
36C2×C4×D173—22no—
37C2×C4×D173—21no—
38C2×C4×D173—21no—
39C2×C4×D173—22no—
40C2×C4×D173—22no—
41C2×C4×D173D412yes—
42C2×C4×D173D412yes—
43#14583803—22no—
44#14583803—22no—
45C4×Dic173—21no—
46C4×Dic173—21no—
47C22×C4×C173C2×C411yescharacteristic
48C22×C4×C173C2×C411yescharacteristic
49C22×C4×C173C2×C411yescharacteristic
50C4⋊C4×C173—22no—
51#13063443—24no—
52#13063443—42no—
53C22×D173—22no—
54C22×D173—22no—
55C22×D173—21no—
56C22×D173—22no—
57C22×D173—21no—
58C22×D173—22no—
59C22×D173C22⋊C412yes—
60C22×D173C2×C812yes—
61C22×D173C2×C812yescomplemented
62C22×D173M1612yes—
63D692—28no—
64C2×Dic173—21no—
65C2×Dic173—22no—
66C2×Dic173C2×D412yes—
67C2×Dic173C2×D411yescharacteristic
68C2×Dic173C2×D411yescharacteristic
69C2×Dic173#4112yes—
70C4×D173—22no—
71C4×D173—24no—
72C4×D173—22no—
73C4×D173—41no—
74C4×D173—42no—
75C4×D173—41no—
76C23×C173C22×C411yescharacteristic
77C2×C4×C173—21no—
78C2×C4×C173—22no—
79C2×C4×C173—21no—
80C2×C4×C173—21no—
81C2×C4×C173—22no—
82C2×C4×C173C22⋊C412yes—
83C2×C4×C173C2×C812yescomplemented
84C22×C173C2×C22⋊C411yescharacteristic
85C22×C173C4×D412yes—
86C22×C173C22×C811yescharacteristic
87C22×C173C2×M1611yescharacteristic
88C22×C173#13112yes—
89C2×D173—22no—
90C2×D173—22no—
91C2×D173—24no—
92C2×D173—24no—
93C2×D173—22no—
94C2×D173—24no—
95C2×D173—41no—
96C2×D173—41no—
97C2×D173—42no—
98C2×D173C22⋊C814yescomplemented
99C4×C173—22no—
100C4×C173—22no—
101C4×C173—41no—
102Dic173—41no—
103Dic173#12011yescharacteristic
104Dic173#12112yes—
105Dic173#12311yescharacteristic
106C2×C173#38611yescharacteristic
107C2×C173C2×C22⋊C811yescharacteristic
108C2×C173#40811yescharacteristic
109C2×C173C8×D412yes—
110C2×C173#43512yes—
111D173—24no—
112D173—42no—
113C173#186911yescharacteristic, complemented, Sylow
114#1869—1731nomaximal, Sylow
115C23×C8—1731no—
116C2×C4×D4—1731no—
117C2×C22⋊C8—1731no—
118C2×C22⋊C8—1732no—
119#336—1732no—
120C22×D4—1731no—
121C23×C4—1731no—
122C23×C4—1731no—
123C22×C8—3462no—
124C22×C8—3464no—
125C22×C8—1732no—
126C22×C8—1732no—
127C4×D4—3462no—
128C4×D4—3462no—
129C2×C4⋊C4#1062192411yescharacteristic
130C2×C22⋊C4—1731no—
131C2×C22⋊C4—1731no—
132C2×C42—1731no—
133C22⋊C8—3462no—
134C22⋊C8—3464no—
135C24—1731no—
136C24—1731no—
137C2×D4—3462no—
138C2×D4—3462no—
139C2×D4—1734no—
140C22×C4—3462no—
141C22×C4—3462no—
142C22×C4—3462no—
143C22×C4—3461no—
144C22×C4—3461no—
145C22×C4—3462no—
146C22×C4—1731no—
147C22×C4—1732no—
148C22×C4—1732no—
149C22×C4—1731no—
150C22×C4C2×#1062192411yescharacteristic
151C22×C4C2×#1062192411yescharacteristic
152C22×C4C2×#1062192411yescharacteristic
153C2×C8—3468no—
154C2×C8—3462no—
155C2×C8—3462no—
156C2×C8—3462no—
157C2×C8—3462no—
158C2×C8—3462no—
159C2×C8—1734no—
160C4⋊C4—22no—
161C22⋊C4—3462no—
162C22⋊C4—3462no—
163C42—3461no—
164C42—3461no—
165C23—3462no—
166C23—3462no—
167C23—3461no—
168C23—3462no—
169C23—3461no—
170C23—3462no—
171C23—1732no—
172C23—1732no—
173C23—1732no—
174C23—1732no—
175C23C22×#1062192411yescharacteristic
176D4—3468no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—3462no—
183C2×C4—3464no—
184C2×C4—3462no—
185C2×C4—3461no—
186C2×C4—3462no—
187C2×C4—1731no—
188C2×C4—1731no—
189C2×C4—1732no—
190C2×C4—1732no—
191C2×C4—6921no—
192C2×C4—6922no—
193C2×C4—6921no—
194C2×C4C22⋊#1062192412yes—
195C2×C4D173⋊C812yescomplemented
196C8—3464no—
197C8—6922no—
198C22—3462no—
199C22—3462no—
200C22—3464no—
201C22—3464no—
202C22—3462no—
203C22—3464no—
204C22—1734no—
205C22—6921no—
206C22—6921no—
207C22—6922no—
208C22C2×C22⋊#1062192411yescharacteristic
209C22D4×#1062192412yes—
210C22C2×D173⋊C811yescharacteristic
211C22D173⋊M1611yescharacteristic
212C22#1914976712yes—
213C4—22no—
214C4—22no—
215C4—41no—
216C4—1731no—
217C4—1732no—
218C4—1731no—
219C4—6921no—
220C2—3464no—
221C2—6922no—
222C2#2601243911yescharacteristic
223C2D173⋊(C22⋊C8)11yescharacteristic
224C2#2601246611yescharacteristic
225C2#2601251912yes—
226C2#2601252112yes—
subgroupG/H
center, FrattiniC23C22×#10621924
commutatorC22×C173C22×C8
FittingC2×C4⋊C4×C173C4
socleC23×C173C22×C4
prime radicalC22×D692C4
cosocleC22×Dic173C23
hypercenterC2×C4⋊C4#10621924
hypocenterC173#1869
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C22×D173⋊C8C2semidirect
2. C2×#19149736C2semidirect
3. C2×#19149736C2semidirect
4. #26012357C2semidirect
27. C2×#13063443D4semidirect
32. C2×D692C8semidirect
61. C22×D173C2×C8semidirect
83. C2×C4×C173C2×C8semidirect
98. C2×D173C22⋊C8semidirect
113. C173#1869semidirect
195. C2×C4D173⋊C8semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×D173⋊C82
C2×#191497362
#260123572
C2×#191496412
#1869173